L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 1.5i)3-s + (0.499 − 0.866i)4-s + 1.73i·6-s + (2.5 − 0.866i)7-s + 0.999i·8-s + (−1.5 − 2.59i)9-s + (2.59 + 1.5i)11-s + (−0.866 − 1.49i)12-s − 3.46i·13-s + (−1.73 + 2i)14-s + (−0.5 − 0.866i)16-s + (−1.73 + 3i)17-s + (2.59 + 1.5i)18-s + (3 − 1.73i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.499 − 0.866i)3-s + (0.249 − 0.433i)4-s + 0.707i·6-s + (0.944 − 0.327i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.783 + 0.452i)11-s + (−0.250 − 0.433i)12-s − 0.960i·13-s + (−0.462 + 0.534i)14-s + (−0.125 − 0.216i)16-s + (−0.420 + 0.727i)17-s + (0.612 + 0.353i)18-s + (0.688 − 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.602019713\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.602019713\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 11 | \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 - 3i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-3.46 - 6i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.79 + 4.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 1.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8.66T + 83T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5.19iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.469656207878715819997858053715, −8.783542092197257976376990217381, −8.080252015827680765578048367885, −7.35207084102659972576620542818, −6.72857943478853574542720227648, −5.70067034991730641933788099789, −4.60084745486195116532359955963, −3.23696899199339005924146749590, −1.92987746647320402307394201218, −0.937412585598720680449585110134,
1.49347120646436122561279592903, 2.67320619240181425819304494532, 3.76150138245593176567265688317, 4.66311804580242315865982378612, 5.61820420054383175827382202207, 6.96531996215229926840083830419, 7.80993614539875141569622901571, 8.821720723594655836674578510837, 9.070966985462560122352804426933, 9.900041916056722761907757525748